On nonlocal symmetries for the Krichever–Novikov equation

Abstract

We construct new infinite hierarchies of nonlocal symmetries and cosymmetries for the Krichever-Novikov equation using the inverse of the fourth-order recursion operator of the latter. PACS 02.30.Ik, 02.30.Jr Keywords: Krichever-Novikov equation, nonlocal symmetries, recursion operatorsThe Krichever-Novikov (KN) equationwhere P (u) = u 3 + c 1 u + c 0 is a third-order polynomial in the reduced form (i.e., without quadratic term and with the leading coefficient equal to 1), c 0 , c 1 ∈ R, has first appeared in [9] in connection with the study of finite-gap solutions of the Kadomtsev-Petviashvili equation which has plenty of physical applications from plasma physics to fluid dynamics, see e.g. [16] and references therein. Eq.(1) is integrable in virtually any reasonable sense: it has infinite hierarchies of local higher symmetries, conservation laws, Hamiltonian and symplectic structures, see e.g. [4]; it also possesses a zero-curvature representation (see e.g. [9,8] and references therein) and a plethora of multisoliton and finite-gap solutions [9,16]. The KN equation is the simplest known integrable one-field elliptic model, see [4].It is readily seen that using suitable fractional linear changes of the dependent variable u we can easily turn (1) into the other known forms of the KN equation with P being a general third-or fourth-degree polynomial in u, cf. [4].It is well known, see e.g. [7,11], that the family of partial differential equations of the formincludes inter alia three fundamental integrable equations, namely, the linear equationthe Korteweg-de Vries (KdV) equationand the KN equation (1). In fact, it is conjectured, see e.g. [7,11] and references therein, that all integrable equations in the family (2) are related via either a classical (point or contact) transformation or a differential substitution, to the above three fundamental integrable equations. Furthermore, this conjecture is proved, see [11] and references therein, for a subfamily of (2) of the formThe symmetry G of (2) by definition (see e.g. [2,3,20] and references therein) satisfies the equationHere and below we identify the symmetry with its characteristic (cf. e.g. [2,20]) and use the so-called Fréchet (or directional) derivative of F ,where u kx = ∂ k u/∂x k and u 0x = u;respectively denote total derivatives with respect to x and t by virtue of (2), cf. e.g. [20]. In addition to local variables x, t, u, u kx the symmetry in (3) may depend on nonlocal variables in the sense of [3] (geometrically, the latter are local coordinates in the fibre of a covering over (2)); roughly speaking, the nonlocal variables are understood as pseudopotentials for (2)). If this is the case, it is tacitly assumed that the total derivatives (e.g. in (3)) are properly extended to the nonlocal variables. Note that the nonlocal symmetries in this sense, i.e., solutions of (3) depending on nonlocal variables, are referred to as shadows of nonlocal symmetries in the terminology of [3].The usual way to construct new symmetries from a known symmetry G is to act on...